Plane Cosserat solids are introduced. It is shown that only one parameter is needed to describe intrinsic rotation. The field equations are simple enough to study the propagation of nonlinear waves and can be reduced to some particular form that admits different nonlinear waves, including solitons and cnoidal waves. We took into account both the matrix–grains and grain–grain interactions, nonlinearity, dispersion, and dissipation. Further examples can be provided of different kinds of analytical approaches, like asymptotical analysis, reduction to the Weierstrass equation, hierarchy of leading equations and waves. Solitary wave solutions and periodic bounded solutions are explicitly obtained.
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